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Asymptotic behavior on the Hénon equation with supercritical exponent

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Abstract

The main purpose of this paper is to analyze the asymptotic behavior of the radial solution of Hénon equation −Δu = |x|α u p−1, u > 0, x ∈ B R (0) ⊂ ℝn (n ⩾ 3), u = 0, x ∈ ∂B R (0), where \( p \to p(\alpha ) = \frac{{2(n + \alpha )}} {{n - 2}} \) from left side, α > 0.

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References

  1. Hénon M. Numerical experiments on the stability of spherica stellar systems. Astronom Astrophys, 24: 229–238 (1973)

    Google Scholar 

  2. Ni W M. A nonlinear Dirichlet problem on the unit ball and its applications. Indiana Univ Math J, 6: 801–807 (1982)

    Article  Google Scholar 

  3. Chen G, Ni W M, Zhou J. Algorithms and visualization for solutions of nonlinear elliptic equations. Internat J Bifur Chaos Appl Sci Engrg, 10: 1565–1612 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  4. Smets D, Su J, Willem M. Non-radial ground states for the Hénon equation. Comm Contemp Math, 4: 467–480 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  5. Cao D, Peng S. The asymptotic behaviour of the ground state solutions for Hénon equation. J Math Anal Appl, 278: 1–17 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  6. Esposito P, Pistoia A, Wei J. Concentrating solutions for the Hénon equation in ℝ2. J Anal Math, 100: 249–280 (2006)

    Article  MathSciNet  Google Scholar 

  7. Peng S. Multiple boundary concentrating solutions to Dirichlet problem of Hénon equation. Acta Math Appl Sin Engl Ser, 22: 137–162 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  8. Pistoia A, Serra E. Multi-peak solutions for the Hénon equation with slightly subcritical growth. Math Z, 256: 75–97 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  9. Serra E. Non radial positive solutions for the Hénon equation with critical growth. Calc Var Partial Differential Equations, 23: 301–326 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  10. Smets D, Willem M. Partial symmetry and asymptotic behavior for some elliptic variational problems. Calc Var Partial Differential Equations, 18: 57–75 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  11. Byeon J, Wang Z Q. On the Hénon equation, asymptotic profile of ground states, I. Ann Inst H Poincaré Anal Non Linéaire, 23: 803–828 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  12. Byeon J, Wang Z Q. On the Hénon equation: asymptotic profile of ground states, II. J Differential Equations, 216: 78–108 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  13. Cao D, Peng S. Asympotic behavior for elliptic problems with singular coefficient and nearly critical Sobolev growth. Ann Mat Pura Appl (4), 185: 189–205 (2006)

    Article  MathSciNet  Google Scholar 

  14. Lin C S, Ni W M, Takagi I. Large amplitude stationary solutions to a chemotaxis system. J Differential Equations, 72: 1–27 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  15. Atkinson F V, Peletier L A. Elliptic equations with nearly critical growth. J Differential Equations, 70: 349–365 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  16. Merle F, Peletier L V. Asymptotic behavior of positive solutions of elliptic equations with critical and supercritical growth I. the radial case. Arch Ration Mech Anal, 112: 1–19 (1990)

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. School of Mathematics and Statistics, Central China Normal University, Wuhan, 430079, China

    ShuJuan Li & ShuangJie Peng

  2. School of Sciences, Jiangxi Bluesky University, Nanchang, 330098, China

    ShuJuan Li

Authors
  1. ShuJuan Li
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  2. ShuangJie Peng
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Corresponding author

Correspondence to ShuangJie Peng.

Additional information

This work was partially supported by National Natural Science Foundation of China (Grant No. 10631030), the Program for New Century Excellent Talents in University (Grant No. 07-0350), the Key Project of Chinese Ministry of Education (Grant No. 107081) and the Scientific Research Foundation for the Returned Overseas Chinese Scholars, Chinese Ministry of Education.

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Li, S., Peng, S. Asymptotic behavior on the Hénon equation with supercritical exponent. Sci. China Ser. A-Math. 52, 2185–2194 (2009). https://doi.org/10.1007/s11425-009-0094-7

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  • DOI: https://doi.org/10.1007/s11425-009-0094-7

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